Euroepan plate armour offered tremendous protection, even if it wasn't full plate. Meanwhile Japanese armour was mostly lamellar, without shields with parts that looked a bit like skirts (lammelar parts protecting thihgs). What exactly made this armour good at their warfare? Why haven't there been an arms race like in Europe, where armours were improved to be better at protection, while weapons were constantly improved to counter that?
First, a summary and timeline of Japanese arms and armour, and warfare:
Ancient period (until approximately AD900): Armies depended on infantry, often armoured, equipped with polearms (primarily spears) or sword (short, often double-edged, straight) and shield. Soldiers could be full-time professionals, or peasants providing short-term compulsory military service.
Medieval period (approx 900-1400): The key component of armies were samurai, who fought as armoured cavalry, primarily as mounted archers, with sabres (the tachi) as sidearms.
Early Modern period (approx 1400 onwards): Armies became larger, necessitating dependence on infantry, most of whom were equipped with spear or other polearms, or bows (up to the mid-16th century) or muskets (after the mid-16th century). Samurai often continued to fight as armoured cavalry, still fighting as mounted archers at times, but often using a lance as their primary weapon. Infantry often wore lighter armour than mounted samurai. Samurai often fought as infantry (resulting in the adoption of the katana as their main sidearm, rather than the cavalry-oriented tachi).
What exactly made this armour good at their warfare?
It stopped spears and arrows. In particular, it could stop arrows from high draw weight bows - we know this both from literary sources which describe arrows being stopped by armour (and exceptional arrows piercing armour), and from the thickness of the armour. Japanese lamellar armour used lamellae of rawhide (3mm thick or greater, up to about 6mm thick) or iron/steel (typically 0.8-1mm thick; iron/steel lamellae could be of composite iron and steel construction, with a hard steel face and a tough iron backing), and depending on the type of construction, at least 2 or 3 thicknesses were presented to arrows due to the side-to-side overlap, and that would be doubled again where there was up-down overlap. The disadvantage is the weight. Note that these heavy samurai armours were intended for cavalry use, which would make the weight more tolerable. Arm and leg armour was thinner (as in Europe), keeping the overall weight tolerable, at the cost of arm and leg armour no longer being reliably arrow-proof.
These armours provide excellent arrow protection, with the shoulder and thigh pieces providing excellent arrow protection for upper arms and legs.
The lack of shields is due to Medieval samurai fighting as mounted archers, and Early Modern Japanese soldiers fighting as spearmen/pikemen or archers/musketeers. Shields, in the form of pavises or mantlets, were still used, but with almost all soldiers using two-handed weapons as their primary weapon, the hand-held shield was of little use. The use of pavises was discussed by /u/PapaJacky in https://www.reddit.com/r/AskHistorians/comments/3mxlon/why_did_japanese_armies_never_use_shields/ a couple of years ago.
Why haven't there been an arms race like in Europe, where armours were improved to be better at protection, while weapons were constantly improved to counter that?
There was an arms race to improve armour and weapons. The first part of this - the adoption of high-draw-weight bows (comparable in anti-armour performance to the English longbow) and the development of body armour and helmets that could stop arrows from such bows - happened quite early (early in period 2 above). The second part, the adoption of guns and the development of bullet-proof armours (body armour and helmets, but not arm and leg armour), was in the 16th century. This included plate cuirasses. Between those, the performance of bows was limited by human strength, and the arms race had already reached its peak. Missing compared to Europe were mechanically-spanned crossbows, which might have pushed armour to improve in the time between these two arms races.
For a good description of the construction of Japanese body armours, see Trevor Absolon, Samurai Armour: Volume I: The Japanese Cuirass, Bloomsbury, 2017.
For an Edo Period Japanese source (in English translation), see Sakakibara Kozan, The manufacture of armour and helmets in sixteenth century Japan, Holland Press, 1963 (originally published in Japanese c. 1800)
What would make European plate armour less useful in Japan than in Europe?
It would have been just as useful in Japan as in Europe. Notably, with the adoption of guns, the improved protection from bullets available with thick plate cuirasses was useful, but this feature was adopted in Japanese armours anyway.
I see /u/wotan_weevil has pretty much answered the question while I was busy answering another question.
I just want to add that Europe did not always use plate armour. Single-piece shaped metal were hard to make, expensive, and did not become common until the late medieval. For many (most?), the armour was mail (rings), which Japan had, coats of plates or brigandine (small plates attached to fabric), which is not that unlike lamellar armour, or gambeson, which was just very thick clothes.
When guns were imported to Japan, the Japanese not only imported European cuirass designs but sometimes the cuirasses themselves, so they very much recognized the protective qualities of the plate cuirass.